NASA Technical Memorandum 4389

Application of Artificial Neural

Networks to the Design Optimization

of Aerospace Structural Components

Laszlo Berke

Lewis Research Center

Cleveland, Ohio

Surya N. Patnaik

Ohio Aerospace Institute

Brook Park, Ohio

Pappu L. N. Murthy

Lewis Research Center

Cleveland, Ohio

National Aeronautics andSpace Administration

Office of Management

Scientific and Technical

Information Program

1993

https://ntrs.nasa.gov/search.jsp?R=19930012642 2018-02-13T18:33:49+00:00Z

Summary

The application of artificial neural networks to capture

structural design expertise is demonstrated. The principal ad-

vantage of a trained neural network is that it requires trivial

computational c|'fort to produce an acceptable new design.

For the class of problems addressed, the development of a

conventional expert system would be extremely difficult, in

the present effort, a structural optimization code with multiple

nonlinear programming algorithms and an artificial neural

network code NETS were used. A set of optimum designs

for a ring and two aircraft wings for static and dynamic con-

straints were generated by using the optimization codes. The

optimum design data were processed to obtain input and out-

put pairs, which were used to develop a trained artificial neu-

ral network with the code NETS. Optimum designs for new

design conditions wcre predicted by using the trained net-

work. Neural net prediction of optimum designs was found

to be satisfactory for most of the output design parameters.

However, results from the present study indicate that caution

must be exercised to ensure that all design variables are

within selected error bounds.

Introduction

The nervous system from slugs to humans follows the

same basic design: neurons connected to many other neurons

forming a biological neural network. The difference between

extremes, such as slugs with only dozens of neurons and hu-

mans with around billions, is the number and complexity of

the connectivities. Their organization and diversity allow for

the specialization of the various areas of this massively paral-

lel, information-processing, living tissue. The human brain is

the ultimate technology with respect to miniaturization and

processing power. The majority of our neurons and their con-

nections reside in the cerebral cortex, the seat of most of our

intellectual capabilities. The cerebral cortex, in physical

terms, is the size of a six-page newspaper, no more than

1/4 in. thick, and crumpled up for packaging in its protective

hard cover. Other parts of the 3-pound wet tissue perform

hard-wired life support functions, including quick-response

emotions, inherited from our reptilian ancestors. At a critical

number of neurons and their connectivities, awareness and

cognition emerged, beginning with our human ancestors mil-

lions of years ago. All living creatures exhibit instinctive or

some level of cognitive reaction to input, responding to feed-

ing opportunity or engaging in threat avoidance. This cogni-

tive tissue has hard-wired, programmable, and self-organizing

capabilities and it is trainable. It has been the subject of in-

tense studies on its anatomy and physiology to its capabilities

and theway it does what it does. We havelearncd much

about the electrochemical activity that occurs in the ncrw)us

system, but the way in which the measurable physical activi-

ties acquire meaning for us is not known now, and is not

likely lobe known in the foreseeable future. The cerebral

cortex has evolved to perform certain tasks better than others.

Vision, for example, is such that a thousand Cray Y-MP's

would have difficulty modeling the same real-lime fidelity

and perception of meaning. At the same time, this ultimate

technology cannot come close to the arithmetic capabilities

of a credit card sized calculator.

Biotechnology and rapidly advancing computer science

have motiwtled the introduction of increasingly sophisticated

artificial neural network models of intriguing brain functions

both in hardware and in software implcmentalions. Vision,

perception, natural language understanding, classification, as-

sociative memory, learning, and accumulation of expertise

arc some targets of this activity. The artificial neural network

(ANN) research is truly muhidisciplinary, encompassing neu-

robiology, physiology, psychology, medical science, math-

ematics, computer science, and engineering.

As in computer science, advancements in ANN are pro-

grossing at a rapid pace. In current conventional ANN appli-

cations, neurons number only in the hundreds with their

connectivilies limited to a few thousand. These numbers will

approach millions or greater in the near future with enhance-

mcnts in computational technology, promising capabilities

approaching lower order living entities. Neural nel models of

learning and the accumulation of expertise inanarrowdo-

main have found their way intc_ practical applications in

many areas. ANN is being attempted for business applica-

tions as a profit-making tool to perform jobs of loan officers,

tax auditors, and stock market experts. Industrial applica-

tions are growing also.

Neural net literature is diverse; only a small sampling can

bc given here (Garret, J.H., Jr. ct at., lt193, J. lntel. Man., to

be published and (refs. 1-4)). Rumelhart and NcCleland (ref.

1) provide a fundamental introduction to the theory of ANN.

The use of a trained neural network as an expert structural

designer was suggested by Bcrkc and Hajcla and is illustrated

at a "toy" problem level (ref. 4). As in structural optimiza-

tion, using mathematical programming techniques, current

neural net capabilities appear to have major limitations in

problem size, especially in the number of variables used in

the mathematical model. The objective of thc investigation

reported here was to further explore the applicability of ANN

when the problem size was computationally feasible for con-

ventional structural optimization.

The expert ANN design model considered here is based on

fced-fl)rward, supervised learning and an error back-propaga-

tion training algorithm. This is the simplest and most popular

ANN paradigm. More sophisticated approaches inw_lving

clustering and classification of dala (ref. 5) or other candidate

approaches, such as functional links (ref. 6) or radial base

functions (RBF), are under investigation at this time. An

ANN is trained first, by utilizing available information gener-

ated from several similar optimum designs of aerospace

structural components. The trained artificial neural network,

as the expert designer, is then used to predict an optimumdesign for a new situation. This situation should resemble

closely, though not identically, the conditions of the training

set, bypassing conventional reanalysis and optimization itera-

tions. The major advantage of a trained neural network as an

expert designer over the traditional computational approachis that results can be produced with trivial computational ef-

fort. Further, the predictive capability of a trained network is

insensitive to numerical instabilities and convergence diffi-

culties typically associated with computational processes

(e.g., during reanalysis, direction generations, one-dimen-

sional searches, and design updates of the nonlinear optimi-

zation schemes). The disadvantage in generating sufficient

design sets to train the artificial neural network is the poten-tial expense.

In this report, the feasibility of ANN as an expert designeris considered for a complex set of engineering problems, rep-

resentative of the optimum designs of structural components

of the aerospace industry. The components are a ring, an in-

termediate complexity wing, and a forward swept wing. The

number of design variables used in the optimization problemsrange from 60 to 157 and the number of behavior constraints

range from 200 to 400. The design load conditions and con-

straint limitations are selected to ensure that, at optimum, allthree types of behavior constraints (i.e., stress, stiffness, and

frequency) become active. The design sets required to train

the neural networks for the three components are generated

with an optimization code CometBoards, which is described

later. The neural network training is carried out through the

code NETS, developed at NASA Johnson Space Center (ref.10). The optimization code CometBoards was run on a Con-

vex mainframe computer at NASA Lewis Research Center to

generate the training data sets, and NETS was run on a SUNSPARC workstation to train the neural networks.

This report is divided into the following five subject areas:

(1) a feed-forward back-propagation artificial neural network,

(2) structural optimization, (3) code CometBoards, (4) dis-

cussion of neural net results, and (5) conclusions.

A Feed-Forward Back-PropagationArtificial Neural Network

Neural network simulations represent attempts to emulate

biological information processing. The fundamcntal proces-

sor is thc neuron, or brain cell, which receives input from

many sources and processes these to generate a unique out-put. The output, in turn,'can be passed on to other neurons.

Learning is accomplished by changing connection strengths

as knowledge is accumulated. The term "neural network"

refers to a collection of neurons, their connections, and the

connection strengths between them. Figure ] shows an ideal-ized neural network where the artificial neurons are shown as

circles, the connections as straight lines, and the connection

strengths (or weights) as calculations derived during the

learning process for a problem. This network contains three

layers-an input layer, an output layer, and a hidden layer-

with each layer consisting of several neurons or nodes. The

adaptation scheme used is based on the popular delta error

back-propagation algorithm. In error back-propagation, theweights are modified to perform a steepest-descent reduction

of the sum of the squares of the differences between the gen-erated outputs and the desired outputs as indicated in the

training pairs.

The optimal number of nodes in the hidden layer and theoptimal number of hidden layers can be problem dependent.

These numbers, however, should be kept low for computa-

tional efficiency. A rule of thumb is to start with a single hid-den layer with the number of nodes equal to about half the to-

tal number of variables in the input and output layers. Thenumbcr of nodes and layers should be increased if conver-gence difficulties are encountered, but should not exceed the

total number of input and output variables. A simpler net-

work with no hidden layers may be computationally efficient,

but it represents only linear mapping between input and out-put quantities. These are known as flat networks and can be

inadequate to model nonlinear relationships. The activationfunction determines the response of the neuron and is the

only source of introducing nonlinearities in the input-outputrelationships.

The details of the back-propagation scheme have been de-

scribed by Rumelhart and NcCleland (ref. 7). A brief discus-

sion of the theoretical background follows.

A typical neural net configuration consists of an input

layer, an output layer, and one hidden layer (as shown in fig.1). Each layer consists of several nodes or neurons. To gain

_ layer

layer

Input

layer

Figure1.--A simpleneuralnetworkmodel.

x- "LJ

Figure 2.--A single processing element,

I=-z

insight into the mechanism of information processing, it is

better to focus on a single node (fig. 2), which receives a set

ofn inputs xi, i = 1,2 .... , n. Thcse inputs arc analogous to

electrochemical signals received by neurons in mammalian

brains. In the simplest model, these input signals are multi-

plied by the connection weights wi), and the effective inputto the elements is the weighted sum of the inputs as follows:

/I

Zj = Z wq x ii=1

(1)

In the biological system, a typical neuron may only pro-

duce an output signal if the incoming signal builds up to a

certain level. This biological characterstic is simulated in the

artificial neural network by processing the weighted sum of

the inputs through an activation function F to obtain an out-

put signal as follows:

V = F(Z) (2)

The type of activation function that was used in the present

study is a sigmoid function. The sigmoid function is given

by the expression

1

F (Z) = 1+ e_,Z+T,, , (3)

This expression was adopted by the NETS computer code

that was used in the present study. In equation (3), Z is the

weighted input to the node, and T is a bias parameter used to

modulate the element output. The principal advantage of the

sigmoid function is its ability to handle both large and small

input signals. The determination of the proper weightcoefficients and bias parameters is embodied in the network

learning process, which is essentially an error minimization

problem.

In the delta error back-propagation approach, the nodes are

initialized arbitrarily with random weights. The output ob-

tained from the network is compared to the actual output (su-

pervised learning) and the error Ei is computed as follows:

E i = (T/-Y/) (4)

where Ti and Yi are the target and the actual output for

node i, respectively. The error signal in equation (4) is mul-

tiplied by the derivative of the activation function for the

neuron in question to obtain

¢5i,k = Ei (5)

where the subscripts i and k denote the i th neuron in the

output layer k. The derivative of the output Yi of the sig-moid function is obtained as follows:

--' = V_(1-_) (6)5Z

The strength of connections between all neurons in the pre-ceding hidden layer to the ira neuron in the output layer is

adjusted by an amount Awpi,k as follows:

AW pi, k = rlSi, kYp, ) (7)

In equation (7), Yp.j denotes the output of neuron p in the

hidden layer j immediately before the output layer; Avvpi,k isthe change in value of the weight between neuron p in the

hidden layer to neuron i in output layer k; and 1/ denotes a

learning rate coefficient (usually selected between 0.01 and

0.9). This learning rate coefficient is analogous to the step

size parameter in a numerical optimization algorithm.

Rumelhart and NcCleland (ref. 1) present a modification to

the approach by including a momentum term as follows:

Awt+lpi, k = ll¢_i,kYp, j + O_Awtpi, k (s)

Superscript t denotes the cycle of weight modification. The

inclusion of the oe term, which would incorporate a memory

in the learning process, increases the stability of the scheme

and helps in preventing convergence to a local optimum.

This approach is applicable, with some variations, in the

modification of weights in other hidden layers. The output of

hidden layers cannot be compared to a known output to ob-

tain an error term. Hence, the following procedure is used.

The _'s for each neuron in the output layer are first com-

puted as in equation (5) and used to determine the weights of

connections from the previous layer. These 5's and w's are

used to generate the 6's for the hidden layer immediately

preceding the output layer as follows:

aP'J = YP'J Z ai'kWpi'k )i(9)

where 6r, j is the 6corresponding to the pth neuron in the

hidden layer and Yp,j is the derivative of the activation func-tion of this neuron as computed in equation (6). Once the

6's corresponding to this hidden layer are obtained, the

weight of connections of the next hidden layer can be modi-

fied by an application of cquation (7) with appropriatechange in the indices. This process is then repeated for all

remaining hidden layers. The process must bc repeated for

all input training patterns until the desired level of error isattained. A modification of the approach is to present a sum

of errors of all training patterns from the very beginning.

Structural Optimization

The structural design optimization problem can be de-

scribed as the following:

Findthc n dcsignvariablcs,_withinprcscribedupper

and lower bounds, (zti "<<_Zi_<Z U, i=1,2 ..... n), which

make the scalar objective function, f(x-), an extremum (a

minimum) subject to a set ofm inequality constraints, rep-

resenting the failure modes of Ihe design problem

gj(_)<_ O, (j = 1,2 ..... m) (lO)

The constraints for structural design applications are typi-

cally nonlinear in the variables _. Equality constraints could

also be included, but generally do not occur in structural de-

sign problems. Behavior parameters considered are stress,

displacement, and frequency constraints g) under multipleload conditions. For each load condition, the stress con-

straints are specified by

ajgj = _-l,O<-O

'-'jo(11)

where crj is the design stress for the jth element and _rjo isthe permissible stress for the jib element. For each load con-

dition, the displacement constraints are specified by

(12)

where u) is the jm displacement component, u)o is the dis-placement limit for the jth displacement component, and jsis the total number of stress constraints. Constraints on fre-

quencies arc specified by

--)

{)<0_:/: f.)(13)

where fn represents natural frequencies of the structure and

fno represents the limitations on these frequencies.

In a mathematical programming technique, the optimal

design point _pt is reached starting from an initial design Zo

in, say, K iterations. The design is updated at each iterationby the calculation of two quantities, a direction 0vector, and

a step length a. The design process can be symbolized as

K

k=l

(]4)

where _k is the direction vector at the k th iteration, and a/:

is the step length along the direction vector 0-k- At the k th

iteration, the direction vector 0k is generated from the

gradients of the objective function and the active constraint

subset following one of the available direction generation

algorithms. Along the direction vector _-k a one-dimensional

search is carried out to obtain the step length cvk, again

utilizing one of several available procedures. The updated

design is then checked against one or more stop criteria and

theiterativcproccss is rcpcated until it convcrges. The

details of the nonlinear mathematical programming

techniques, although well documented in the literature, arenot elaborated here.

Code CometBoards

The basic structure of the optimization code ComctBoards

(Berke, L., Guptill, J., and Patnaik, S.N., 1993, NASA TP, to

be published) is depicted in figure 3. The code has a centralcommand unit, control via command level interface, as

shown in figure 3. This unit establishes links between the

three modules of the code (optimizer, analyzer, and data files)

to solve the optimization problem. The solution is stored in

an output device. Scvcral options for optimizers and analyz-ers are available. The optimization options are

(1) Fully utilized design (FUD)

(2) Optimality criteria (OC) technique

(3) Method of feasible directions (FD)

(4) International Mathematical and Scientific Library

(IMSL) sequential quadratic programming (SQP)method

(5) Sequential linear programming (SLP)

(6) Sequential quadratic programming (SQP)

(7) Sequential unconstrained minimization technique

(SUMT)

The analyzer options are the displacement method, the inte-

grated force method, and the simplified force method.

There are three input data files: ANLDAT, DISDAT, and

OPTDAT. A typical command to execute the code Comet-Boards is

Optimize SUMT disp other stress disp freq ( Output sdf a

CometBoards

Comparative Evaluation lest Bed of

Optimizers and _nalyze[s

for the Design of Structures

Optimization options

• FUD• OC Analyzer options Data files

• FD * Displacement • Demo problems (9)• IMSL (sqp) • Force • User generated• SLP • Others (Berke)• SQP• SUMT

ISDAT)(OPTDAT)

_Control via command/

level interface /

l • Iris/Unix (C Shell Script)• VM/CMS (REXX Exec)

[ Results I

• Stored in file

• Displayed at terminal

Figure 3.--Optimization code CometBoards.

followed by three data files (prompted interactively),

ANLDAT filcl a, D1SDAT filcl a, and SUMTDAT filel a.

The first two arguments "Optimize SUMT" represent optimi-

zation using SUMT. The third argumcnt "disp" mcans dis-

placemcnt method will be used as thc analysis tool. The

flmrth argument "other" is a name for the optimization prob-

lem. The fifth, sixth, and seventh arguments "stress," "disp,"

and "freq" indicate the types of behavior constraints consid-

ered: stress for stress constraints, disp for displacement, and

freq for frequency constraints.

The file ANLDAT filel a is the analysis input data file

from which the finite clement analysis information is read.

The file D1SDAT filel a is the design input data file from

which information required to sct up the optimization

problem is read. The filc SUMTDAT filel a is the

optimization input data file for optimizcr SUMT. Results of

the optimization problem are stored in the filc Output sdf a.

In brief, thc code CometBoards has considcrable flexibility in

solving a design problem by choosing one of several

optimizers and onc of three analysis mcthods. Space does

not permit further discussion of ComctBoards. However, the

code is used to gencrate sets of optimum designs for the ring,

the intermediate complexity wing, and the forward swept

wingproblcms used to train thc network. Tocnsurcthc

reliability of the optimum designs, the same problcms were

solved using several differcnt optimizers and analyzers.

Discussion of Neural Net Results

The prcdictivc capabilities of the trained ncural networks

as cxpcrt designers are described for all thrcc examples in

this section. The ANN training's predictive capability for the

ring problem is described in some detail; however, only

cursory discussion will be included hcrc fl_r the other wing

problems.

Example I --The Trussed Ring

The trussed ring (fig. 4) is the first example to illustrate lhc

predictive capabilities of ANN. The inner and outer radii of

the ring arc R i and R,, rcspcctivcly. The ring is idealized

by 60 truss elements madc of aluminum with Young's

modulus, E = l0 000 ksi and weight density p=0.1 lb/in. 3.

The design of the ring for minimum weight under stress,

displacement, and frequency constraints was used for the

artificial neural nctwork training. The ring was subjected to

three static load conditions as given intahlc 1. Lumped

4

S ,,1

Figure 4.--Base configuration for trussed ring.

TABLE 1.- LOAD SPECIFICATIONS FOR 60-

BAR TRUSSED RING

I

II

III

_d condilions

Node number

I

7

15

18

22

rnped masses 4

12

Load components, kip

Px Py

--10 0

9 I)

-8 3

-8 3

-20 I(1

m =m =21J0 Ib

rn =mr=2lRl Ib

TABLE II. -- CONSTRAINT SPECIFICATION FOR

60-BAR TRUSSED RING

Constrain! type Constraint descriptiont

'lStress;,' tri<tro(i=l ,2, ... ,60)

o'o= 1(1 ksi

! Displaccmcnl Magnitudc in direclions x and y

Node number, i Ui,in.

4 1.75

I(I 1.25

13 2.25

19 2.75

Frcqucncy f > fo

f,=13, 14, 15, 16, or 17 Hz

Design hounds for Ai>O,5 in.2(i=1,2,,..,61))3rc_lN

TABLE 111.-- DESIGN VARIABLE

LINKAGE FOR 60-BAR

TRUSSED RING

Dcsign variablc

IO

1t

12

13

14

15

16

17

18

19

2(1

21

22

23

24

25

Members linked

I 49 through 60

2 1,13

3 2,14

4 3,15

5 4,16

6 5,17

7 6,18

8 7,19

9 8,20

9.21

10,22

11,23

12,24

25,37

26,38

27,3q

28,4()

29,4 I

3(I,42

31,43

32,44

33,45

34,46

35,47

36,48

masses were used for frequency calculations, whereas the

elements were idealized as massless springs. The constraint

specifications are given in table 11. The optimum design of

the ring was determined using a total of 195 behavior

constraints, consisting of 180 stress, 24 displacement, and 1

natural frequency constraint. The 60-bar cross-sectional

areas shown in table Ill were linked to obtain a reduced set of

25 design variables. The values for loads, masses, and

bounds for displacement and frequencies were specified to

ensure that, at optimum, the active set included all three types

of constraints (i.e., stress, displacement, and frequency). For

neural network simulation, the geometry of the ring is

controlled by its inner and outer radii. These radii and the

frequency limits are the three global input parameters. The

25 linked design parameters and the minimum weight make

up the 26 output variables. For neural net training and

predictions, 125 sets of optimum designs are generated using

all combinations of 5 outer radii (Ro=80, 90, 100, 110, and

120 in.), and 5 inner radii (R i = 0.92Ro, 0.91R o, 0.90R o,

0.89Ro, and 0.88Ro), along with 5 frequency limits (13, 14,

15, 16, and 17 Hz).

Figure 5 shows the composite design configuration for the

trussed ring. The optimum designs were obtained using

sequential unconstrained minimization techniques (SUMT).

The optimum design convergence characteristics were

verified by solving the problem using two other methods:

Figure 5._Composite design configuration for trussed ring.

(1) method of feasible directions (FD) and (2) sequential

quadratic programming (SQP) techniques. The optimum

weights of the 125 designs varied from 1 000 lb to

approximately 150 000 lb. At optimum, frequency is

typically an active constraint along with other stress and

displacement constraints. Simply stated, the design is

complex with a highly undulated design space. TheCPU

time in a Convex machine to generate one optimum design of

the ring using CometBoards can take between 3 to 45 rain

depending on the optimizer and its convergence parameters

as well as on the analyzer and reanalysis schemes. Fifteen

minutes of CPU time in a Convex machine can be considered

typical for the solution of a ring optimization problem.

However, the generation of 125 optimum designs required

more than 35.125 CPU hr because of convergence difficulties

encountered during optimization runs.

For the purpose of training and prediction of optimum

structural design for stress, displacement, and frequency con-

straints through an artificial neural network, the 125 data sets

were separated into a training set consisting of 120 designs

and a test set of 5 designs. These input and output pairs were

used to train a back-propagation neural network using the

code NETS. The trained network was then used to predict

the design for the test set.

The parameters specified to train the ring design by using

the ANN code NETS in a SUN SPARC workstation were (1)

the number of hidden layers, one with 30 nodes (Two hidden

layers with 15 nodes each were also used with othcr

parameters unchanged, but no significant training efficiency

could be attributed to such variations.) and (2) the root mcan

square error at 1/10 of 1 percent with a learning rate of 0.9

and training iterations of 10 000 cycles. The SUN SPARe

workstation required about 10 hr to complete the 10 000

training cycles. At this training level, appreciable errors in

both training and test sets were observed. For superior

convergence of the weights, the ANN training was

augmented by another 10 000 cycles with the total training

30Test set

•_ NNI 1> 20 _ 2

05 15

Error groups, percent

Figure 6._Error histogram for three test data sets fortrussed ring.

25

cycles at 2(} 000. At intermediate training steps, the learning

rate was progressively reduced from 0.9 to 0.1. Weights,

checked at intermediate training cycles, remained stationary

with overall rms error at 0.029 at the end of 10 000 cycles

and 0.025 at the end of 20 000 training cycles.

The NETS simulation results indicated that the training

process was satisfactory for most of the variables.

Predictions displayed individual error rates between 0 to 10

percent for approximately 80 percent of the variables.

However, a few variables exhibited a much higher incidence

of error. The test set followed the pattern of the training set

with minor differences. Considering the complexity of the

optimum designs (shift of load-carrying paths and weight

range of I 000 to 150 000 Ib) the training and predictions

cannot be considered unsatisfactory because even a veteran

designer would have experienced difficulty in estimating forthe situation.

To improve the performance of ANN predictions, the

complexity of the optimum designs and associated

undulations of the design space was reduced to a certain

extent bv considering only 34 data sets out of the 125

dcsigns. The optimum weights of these 34 designs varied

between I /R_0 and 5 (,R)0 lb. The 34 data sets were separated

>Figure7.--Base configurationfor intermediatecomplex-

itywing.

into a training set of 31 data sets and 3 test data sets. As

before, these input and output pairs were used to train the

same back-propagation network, which was then used to

predict the design for the test sets. The training parameters

used were identical to the ones used previously.

As expected, substantial improvement in accuracy was

noticed for this training set. Training set predictions

exhibited reduced error between 0 to 5 percent for most of

the variables with some exceptions. The test set followed thepattern of the training set with minor differences. The

performance of the neural network to predict the ring design

for three test data sets is shown in figure 6 in the fi)rm of an

error histogram. Here the neural net output was compared

with the target values and errors were computed for eachoutput variable. These errors were then grouped according to

their magnitudes and plotted in the form of a histogram. The

procedure was repeated for each test data sct. Thus, for

example, in figure 6 it can be seen that all 26 output variableswere predicted within 5 percent for the test data set 2, The

other two data sets exhibited error in 10-, 15-, and 20-percent

groups for a few output variables. Nole that an error group of15 percent, for example, suggests that all output variables in

this group were predicted with an absolute error in the range

of 10 to 15 percent. Most of the predictions for all three test

data sets exhibited error of approximately 5 percent. A few

variables exhibited crmr exceeding 5 percent, but there wasno variable for which thc incidcncc of error cxcccdcd 25

percent.

Example 2--Optimum Design of an Intermediate

Complexity Wing

The intermediate complexity wing represents the second

illustrative example. Figure 7 shows the geometrical

configuration of the wing. The approximate length of the

wing is 90 in. Its width at the base is about 48 in. and about

29 in. at the apex. Its depth varies between 2.25 in. at the

base to 1.125 in. at the apex. The finite clement modcl has

88 grid points and a total of i58 elemcnts consisting of 39

Figure8.--Composite designconfigurationfor intermediatecomplexitywing.

3fitxl

.t-in

e'-c71

-8"6

E.-iZ

60 -

°°I2O

05

r] fl n

15 25 35 45 55 65

Error groups, percent

75 85 95

Figure 9.--Error histogram for several trained data setsfor intermediate complexity wing.

bars, 2 triangular membranes, 62 quadrilateral membranes,

and 55 shear panels. The structure is made of aluminum with

Young's modulus E = 10 500 ksi, Poisson's ratio v=(1.3, and

weightdcnsityp=(I. 1 Ib/in. 3. Thc design of the wing for

minimum weight, under displacement constraints specified at

the apex of the wing, was used for the artificial neural

network calculations. The 158 design variables consisting of

bar areas and plate thicknesses were linked to obtain a

reduced set of 57 design parameters for optimization. For

neural network simulation, the geometry of the wing was

changed to approximatcly 1.25 times its plan area. The basic

design and the composite-perturbed configurations are shown

in figures 7 and 8, respectively. Each geometrical

configuration was specified by a single master parameter as

the input variable. The 57 optimum design parameters and

the minimum weight are considcred thc 58 output

parameters. Fiftccn sets of optimum designs are generated

for 15 different geometrical configurations. The optimum

designs were obtained by using sequential unconstrained

minimization techniques (SUMT), which were verified

further by two other optimizers: (1) method offcasible

directions (FD) and (2) sequential quadratic programming

(SOP) techniques. The influcnce of compatibility or

indeterminacy, which changes the load paths in the structure

for different optimum designs, was observed among the 15

data sets. The 15 data sets were separated into a training set

consisting of 13 data sets and a test set of 2 data sets to

predict the optimum structural design of the wing by using an

artificial neural network. These input and output pairs were

used lo train a neural network. The ANN paramcterswere

kept identical to those of the ring problem. Figure g shows

thc results for the wing in an error histogram form. This

histogram shows that most predictions exhibited error of

about 5 percent. There are a few variables that strayed into

higher error brackets.

Figure 10.---Base configuration for forward swept wing.

Example 3--The Forward Swept Wing

The forward swept wing (fig. 1(1) reprcscnts the final

illustrative example. The approximate length of the wing is

160 in. Its width at the base is about 8,0 in. and about 40 in.

at the apex. Its depth varies between 2(1 in. at the base to 10

in. at the apex. The finite clement model h;ls 3(1 grid points

and 135 truss elements. The structure is made of aluminum

with Young's modulus E=IO 0()0 ksi, F'oisson's ratio 1,=0.3,

and weight densityp=0.11b/in. 3. Thc optimum design of thc

wing for minimum weight, under displacement constraints at

its apex, was used for the artificial neural network

Figure 11 .--Composite design configuration for forward swept wing.

'C

>C

'3

EZ

6O0

_ n

05 15 25

Errorgroups,percent

Figure12.---Errorhistogramfor four datasetsforforward sweptwing.

calculations. All 135 member areas were regarded as

independent design variables. For neural network simulation,

the geometry of the wing was changed to approximately 1.5

times its base line plan area. Figure 11 shows the composite

configuration of the forward swept wing. Each 1-percent

change in its configuration was considered one data set for

the training of the artificial neural network. Fifty optimum

designs were obtained for the 50 geometrical configurations.The 50 designs were separated into a training set consisting

of 45 cases and a test set consisting of the remaining 5 sets.

As in the previous problems, a back-propagation neural

network was trained by using NETS with training parameters

identical to the previous cases. Figure 12 presents the results

in bistogram form. Most predictions exhibited about 5-

percent error. However, a few variables exhibited 15- to 25-

percent error, as shown in the error historgram for four testdata sets.

Conclusions

Artificial neural network predictions of optimum designs

under difficult design requirements were found to be satisfac-

tory for most of the output design parameters. A few design

parameters, however, strayed into higher error brackets.The errors can be reduced to some extent when the com-

plexity of the design space is reduced by using a much nar-

rower design range, or by increasing the number of training

sets to provide a better supervised learning environment for

the neural network. Within the context of structural design

data, caution should be exercised because load paths can

change as a result of the indeterminacy of the structure.

Discontinuities can be created in the design space that are

difficult to model with the simple ANN code. Techniques are

being made available for clustering the training data that rep-resent similar behavior and then decomposing the network

for training within the clusters. Training data preprocessing

and other training paradigms, such as radial base functions

(RBF), are worth exploring because they show potential to

increase the robustness necessary for a neural network to act

as an expert designer.

More research is required to assess the viability and

usefulness of a neural network as an expert designer in

routine design applications. The power of a trained ANN is

in its capability to generalize and in its instantaneous

response regardless of the original complexity of the

problem. A trained ANN can provide very good estimates

for optimum designs for what-if situations under changing

conditions and at trivial computing cost. Such estimates can

be used, not only in the structural problem setting, but in a

multidisciplinary environment because the calculation ofsensitivities becomes trivial once the network is trained.

There are many other intriguing possibilities, particularly if

one considers the rapidly expanding ANN capabilities and

improvements in associated hardwares. These factors will

open applications by reducing the training effort to

acceptable computational costs even for much larger

problems than those illustrated here.

Lewis Research Center

National Aeronautics and Space Administration

Cleveland, Ohio, July 1, 1992

References

1. Rumelhart, D.E.; and NcCIcland, J,L.: Parallel Distributed Processing:

Explorations in the Microstructure of Cognition, Vols. l&2. The

MIT Press, Cambridge, MA, [_188.

2. Bahr, B.; and Nabcel, T.M: Neural Networks for Delecting Defects in

Aircraft Structures. IAR Report 90-4, Institute for Aviation

Research, Thc Wichita State Universily, KS, 1990.

3. Tank, D.W.; and Hopficld, .I.J.: Simple "Neural" Optimization Net-

works: An A/D Converter, SignaI Decision Circuit, anda Linear

Programming Circuit. IEEETrans. CircuitsSyst.,voI. CAS-33,

1986, pp. 533-541.

4. Berkc, L.; and Hajela, P.: Application of Artificial Neural Nets in

Structural Mechanics. NASA TM-102420, 1990.

5. Hajela, P.; Fu, B.; and Berke, L.: ART Networks in Automated

Conccptual Design of Structural Systems. Prcsentcd at the Second

International Conference on the Application of Artificial Intelligence

Techniques to Civil and Structural Engincering, Oxford, England,

Sept. 3-5, 199 I.

6. Pao, Y.H.: Adaptive Pattern Recognition and Neural Networks.

Addision-Wcsley Publishing Co., Reading, MA, 1989.

7. Ming, G.; and Xila, L.: A Preliminary Design Expert System (SPRED-

1) Based on Neural Networks. Presented at the Second International

Conference on the Application of Artificial Intelligence Techniques

to Civil and Structural Engineering, Oxford, England, Scpt. 3-5,

1991.

8. Brown, D.A.; Murthy, P.L.N.; and Berkc, L.: Computational Simula-

tion of Composite Ply Micromcchanics Using Artificial with Neural

Networks. Microcomput. Civil Eng., vol. 6, 1991, pp. 87-97.

9. Swift, R.A.; and Batill, S.M.: Application of Neural Nctworks to

Preliminary Structural Design. AIAA/ASME/ASCE/AHS/ASC

Slruclures, Struelural Dynamics and Materials Conference, 32nd,

AIAA, New York, vol. 1, 1991, pp. 335-343.

10. Baffes, P.T.: NETS 2.0 Users Guide. LSC-23366, NASA Lyndon B.

Johnson Spacc Center, Sept., 1989.

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Application of -\rtificial Neural Networks to lhe Design Optimization of

Aerospace Slructural Components

6. AUTHOR(S)

Laszlo Berke, Surva N. Patnaik. and Pappu L.N. Murthv

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS{ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

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National Aeronautics and Space Administration

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Laszlo Berke and Pappu L. Murthy, NASA Lewis Research Center, Cleveland, Ohio, and Surya N. Pamaik, Ohio

Aerospace lnstitute, 2001 Aerospace Parkway', Brook Park, Ohio44142. Responsible person, Surya N. Patnaik

(21_)433-8468.

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13. ABSTRACT (Maximum 200 words)

The application of artificial neural networks to capture structural design expertise is demonstrated. The principal

advantage of a trained neural network is that it requires trivial computational effort to produce an acceptable new design.

For the class of problems addressed, the development of a conventional expert system would be extremely difficult. In

the present effort, a structural optimization code with multiple nonlinear programming algorithms and an artificial neural

network code NETS were used. A set of optimum designs for a ring and two aircraft wings for static and dynamic

constraints were generated by using the optimization codes. The optimum design data were processed to obtain input and

output pairs, which were used to develop a trained artificial neural network with the code NETS. Optimum designs for

new design conditions were predicted by using the trained network. Neural net prediction of optimum designs was found

to be satisfactory for most of the output design parameters. However, results from the present study indicate that caution

must be exercised to ensure that all design variables are within selected error bounds.

14. SUBJECT TERMS

Artificial neural network; Structures; Design; Optimization; Experl designer

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